You need to split jobs into consecutive segments and minimize the sum of each segment's maximum. This is a partitioning DP.
Define dp[day][i] as the minimum total difficulty to schedule jobs through using exactly day days. On day , there's no choice: dp[1][i] is the maximum of the first jobs.
For later days, decide where to place the cut. If day day covers jobs through , then dp[day][i] = dp[day-1][j-1] + max(jobs[j..i]). Try each valid and take the minimum.
Each day needs at least job, so the earliest starting index on day day is day - 1. This prunes invalid states and keeps the recurrence well-defined.